where \(m(\vec{x},t)\)  denotes the local magnetization field and \(\eta \)  is an uncorrelated white noise term, is an effective coarse-grained model, well-known to describe the evolution of the scalar magnetization of an Ising ferromagnet around thermal equilibrium [1]}, [2]}. We use this model in order to simulate the full dynamic evolution of our Ising system. We also define here a coarse-grained spin lattice \(\lbrace  s_{\vec{r}} \rbrace \)  by discretizing \(s_{\vec{r}}=+1\)  if \(m(\vec{r})>0\)  and \(s_{\vec{r}}=-1\)  otherwise, from which we will define the field \(h(x,(y))\)  using Eq. (REF ). The same boundary conditions as those proposed in Ref. [3]} are considered, see Fig. REF .
